Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Fiber bundles in quantum physics

Rathindra Nath Sen, Geoffrey L. Sewell

Published Mar 1, 2002• 15 citations
Publisher

Abstract

The theory of fiber bundles provides a natural setting for the description of macroscopic quantum systems, wherein their classical and quantum features are represented by actions on the base manifolds and the fibers, respectively, of the relevant bundles. We provide realizations of this picture in the description of (a) quasiparticle excitations of many-body systems, especially those in superfluid helium, (b) the interplay between the microscopic and macroscopic dynamics in certain irreversible processes, such as that of a laser, and (c) local thermodynamic equilibrium. In particular, (b) involves the treatment of a dynamical system which is defined on a vector bundle.

DOI
10.1063/1.1447309

Related papers

Macroscopic World and Microscopic World

Takao Tati

A physical picture of macroscopic and microscopic worlds is presented. In the space-time description, both the macroscopic and microscopic worlds are described simultaneously in space-time being a single boundless four dimensional continuum in which the motion of the observer is represented as a world line. In our picture, the space-time description is valid only for the macroscopic world and the microscopic world is local in the sense that it is described in a finite space associated with each point of the macroscopic space-time. In this case, space-time does not exist in the microscopic world in which the observer can not move. For the description of our picture, the language of the non-space-time description is necessary in which space-time is not presupposed. We can utilize the space-time description for the macroscopic world and, at the same time, we have a theory of finite degree of freedom describing creations and annihilations of elementary particles in the microscopic world free from the divergence difficulty, instead of the divergent quantum field theory in the space-time description which is considered in our picture a limit of an approximation like the ideal gas in gas theories. The finite degree of freedom is given by a one-particle state-vector space of a finite dimension from which the second quantization leads to a state-vector space of a countable dimension (a Hilbert space) for the theory of creations and annihilations of elementary particles describing a local microscopic world. The relation between microscopic worlds associated with different macroscopic space-time points is given by a dynamical principle up to a phase factor due to possible fluctuations of phases of local state-vector spaces. The experiment on interference of light performed recently by Franson and Potocki is interpreted in terms of fluctuation of the local phases and is considered to support the theory of finite degree of freedom which leads to the idea of local microscopic worlds.

Annals of the Japan Association for Philosophy of Science•Jan 1, 1986•4 citations

QUANTUM BUNDLES AND QUANTUM INTERACTIONS

Daniel Canarutto

A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ("quantum bundles") over the space–time manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a "detector"); in the flat, inertial case this reproduces the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.

arXiv:math-ph/0506058•International Journal of Geometric Methods in Modern Physics•Oct 1, 2005•35 citations

Geometric quantum thermodynamics: A fibre bundle approach

T. Pernambuco, L. C. Céleri

Classical thermodynamics is a theory based on coarse-graining, meaning that the thermodynamic variables arise from discarding information related to the microscopic features of the system at hand. In quantum mechanics, however, where one has a high degree of control over microscopic systems, information theory plays an important role in describing the thermal properties of quantum systems. Recently, a new approach has been proposed in the form of a quantum thermodynamic gauge theory, where the notion of redundant information arises from a group of physically motivated gauge transformations called the thermodynamic group. In this work, we explore the geometrical structure of quantum thermodynamics. Particularly, we do so by explicitly constructing the relevant principal fibre bundle. We then show that there are two distinct (albeit related) geometric structures associated with the gauge theory of quantum thermodynamics. In this way, we express thermodynamics in the same mathematical (geometric) language as the fundamental theories of physics. Finally, we discuss how the geometric and topological properties of these structures may help explain fundamental properties of thermodynamics.

quant-pharXiv:2512.14383•Physical Review A•Dec 17, 2025•1 citation

Superfluidity

C. J. Pethick, H. Smith

The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.

Cambridge University Press eBooks•Nov 29, 2001•57 citations

Fundamentals of Algebraic Quantum Theory

Alfred Rieckers

Many body physics is concerned with macroscopic systems consisting of a very large number of microscopic particles (N ≈ 10 20 ). In this case the direct dynamical treatment of the complete system is impossible. A quite useful and frequently applied approximation consists in the neglection of terms which are relatively small as the volume of the systems tends to infinity, the particle density remaining constant. For the concise theoretical description of the typical many body phenomena such as phase transitions and spontaneous symmetry breakdown the use of the thermodynamical limit is even necessary, since no existing theory for the finite systems shows these effects. As a general rule one can say that a quantum system behaves like a thermodynamical system only in the infinite-volume limit. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vieweg+Teubner eBooks•Jan 1, 1980•2 citations

Geometric foundations of thermodynamics in the quantum regime

Álvaro Tejero, Martín de la Rosa

In this work, we present a geometrical formulation of quantum thermodynamics based on contact geometry and principal fiber bundles. The quantum thermodynamic state space is modeled as a contact manifold, with equilibrium Gibbs states forming a Legendrian submanifold that encodes the fundamental thermodynamic relations. A principal fiber bundle over the manifold of density operators distinguishes the quantum state structure from thermodynamic labels: its fibers represent non-equilibrium configurations, and their unique intersections with the equilibrium submanifold enforce thermodynamic consistency. Quasistatic processes correspond to minimizing geodesics under the Bures-Wasserstein metric, leading to minimal dissipation, while the divergence of geodesic length toward rank-deficient states geometrically derives the unattainability aspect of the third law. Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies a geometric source of irreversibility in cyclic processes. In this framework, the thermodynamic laws in the quantum regime emerge naturally as geometric consequences.

math-phCross-listedarXiv:2511.10125•Journal of Physics A Mathematical and Theoretical•Nov 14, 2025•0 citations
The theory of fiber bundles provides a natural setting for the description of macroscopic quantum systems, wherein their classical and quantum features are represented by actions on the base manifolds and the fibers, respectively, of the relevant bundles. We provide realizations of this picture in the description of (a) quasiparticle excitations of many-body systems, especially those in superfluid helium, (b) the interplay between the microscopic and macroscopic dynamics in certain irreversible processes, such as that of a laser, and (c) local thermodynamic equilibrium. In particular, (b) involves the treatment of a dynamical system which is defined on a vector bundle.
A physical picture of macroscopic and microscopic worlds is presented. In the space-time description, both the macroscopic and microscopic worlds are described simultaneously in space-time being a single boundless four dimensional continuum in which the motion of the observer is represented as a world line. In our picture, the space-time description is valid only for the macroscopic world and the microscopic world is local in the sense that it is described in a finite space associated with each point of the macroscopic space-time. In this case, space-time does not exist in the microscopic world in which the observer can not move. For the description of our picture, the language of the non-space-time description is necessary in which space-time is not presupposed. We can utilize the space-time description for the macroscopic world and, at the same time, we have a theory of finite degree of freedom describing creations and annihilations of elementary particles in the microscopic world free from the divergence difficulty, instead of the divergent quantum field theory in the space-time description which is considered in our picture a limit of an approximation like the ideal gas in gas theories. The finite degree of freedom is given by a one-particle state-vector space of a finite dimension from which the second quantization leads to a state-vector space of a countable dimension (a Hilbert space) for the theory of creations and annihilations of elementary particles describing a local microscopic world. The relation between microscopic worlds associated with different macroscopic space-time points is given by a dynamical principle up to a phase factor due to possible fluctuations of phases of local state-vector spaces. The experiment on interference of light performed recently by Franson and Potocki is interpreted in terms of fluctuation of the local phases and is considered to support the theory of finite degree of freedom which leads to the idea of local microscopic worlds.
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ("quantum bundles") over the space–time manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a "detector"); in the flat, inertial case this reproduces the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.
Classical thermodynamics is a theory based on coarse-graining, meaning that the thermodynamic variables arise from discarding information related to the microscopic features of the system at hand. In quantum mechanics, however, where one has a high degree of control over microscopic systems, information theory plays an important role in describing the thermal properties of quantum systems. Recently, a new approach has been proposed in the form of a quantum thermodynamic gauge theory, where the notion of redundant information arises from a group of physically motivated gauge transformations called the thermodynamic group. In this work, we explore the geometrical structure of quantum thermodynamics. Particularly, we do so by explicitly constructing the relevant principal fibre bundle. We then show that there are two distinct (albeit related) geometric structures associated with the gauge theory of quantum thermodynamics. In this way, we express thermodynamics in the same mathematical (geometric) language as the fundamental theories of physics. Finally, we discuss how the geometric and topological properties of these structures may help explain fundamental properties of thermodynamics.
The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.
Many body physics is concerned with macroscopic systems consisting of a very large number of microscopic particles (N ≈ 10 20 ). In this case the direct dynamical treatment of the complete system is impossible. A quite useful and frequently applied approximation consists in the neglection of terms which are relatively small as the volume of the systems tends to infinity, the particle density remaining constant. For the concise theoretical description of the typical many body phenomena such as phase transitions and spontaneous symmetry breakdown the use of the thermodynamical limit is even necessary, since no existing theory for the finite systems shows these effects. As a general rule one can say that a quantum system behaves like a thermodynamical system only in the infinite-volume limit. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this work, we present a geometrical formulation of quantum thermodynamics based on contact geometry and principal fiber bundles. The quantum thermodynamic state space is modeled as a contact manifold, with equilibrium Gibbs states forming a Legendrian submanifold that encodes the fundamental thermodynamic relations. A principal fiber bundle over the manifold of density operators distinguishes the quantum state structure from thermodynamic labels: its fibers represent non-equilibrium configurations, and their unique intersections with the equilibrium submanifold enforce thermodynamic consistency. Quasistatic processes correspond to minimizing geodesics under the Bures-Wasserstein metric, leading to minimal dissipation, while the divergence of geodesic length toward rank-deficient states geometrically derives the unattainability aspect of the third law. Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies a geometric source of irreversibility in cyclic processes. In this framework, the thermodynamic laws in the quantum regime emerge naturally as geometric consequences.